The Riemann zeta function and tuning: Difference between revisions

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This new function has the property that {{nowrap|Fs(''x'') {{=}} Fs(0) − Es(''x'')}}, so that all we have done is flip the sign of Es(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the [[Wikipedia:Riemann zeta function|Riemann zeta function]]:

<math>\displaystyle F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</math>

<math>\displaystyle F_s(x) = \Re \ln \zeta\left(s + \frac{2 \pi i}{\ln 2}x\right)</math>

If we take exponentials of both sides, then

<math>\displaystyle \exp(F_s(x)) = \abs{\zeta(s + 2 \pi i x/\ln 2)}</math>

<math>\displaystyle \exp(F_s(x)) = \abs{\zeta\left(s + \frac{2 \pi i}{\ln 2}x\right)}</math>

so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.

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 This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) &minus; E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the [[Wikipedia:Riemann zeta function|Riemann zeta function]]:
-<math>\displaystyle F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</math>
+<math>\displaystyle F_s(x) = \Re \ln \zeta\left(s + \frac{2 \pi i}{\ln 2}x\right)</math>
 If we take exponentials of both sides, then
-<math>\displaystyle \exp(F_s(x)) = \abs{\zeta(s + 2 \pi i x/\ln 2)}</math>
+<math>\displaystyle \exp(F_s(x)) = \abs{\zeta\left(s + \frac{2 \pi i}{\ln 2}x\right)}</math>
 so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.